Lemmas about List.range and List.enum

Most of the results are deferred to Data.Init.List.Nat.Range, where more results about natural arithmetic are available.

Ranges and enumeration

range'

theorem List.range'_succ (s n step : Nat) : List.range' s (n + 1) step = s :: List.range' (s + step) n step

@[simp]

theorem List.length_range' (s step n : Nat) : (List.range' s n step).length = n

@[simp]

theorem List.range'_eq_nil {s n step : Nat} : List.range' s n step = [] ↔ n = 0

theorem List.range'_ne_nil (s : Nat) {n : Nat} : List.range' s n ≠ [] ↔ n ≠ 0

@[simp]

theorem List.range'_zero {s step : Nat} : List.range' s 0 step = []

@[simp]

theorem List.range'_one {s step : Nat} : List.range' s 1 step = [s]

@[simp]

theorem List.tail_range' {s step : Nat} (n : Nat) : (List.range' s n step).tail = List.range' (s + step) (n - 1) step

@[simp]

theorem List.range'_inj {s n s' n' : Nat} : List.range' s n = List.range' s' n' ↔ n = n' ∧ (n = 0 ∨ s = s')

theorem List.mem_range' {s step m n : Nat} : m ∈ List.range' s n step ↔ ∃ (i : Nat), i < n ∧ m = s + step * i

theorem List.getElem?_range' (s step : Nat) {m n : Nat} : m < n → (List.range' s n step)[m]? = some (s + step * m)

@[simp]

theorem List.getElem_range' {n m step : Nat} (i : Nat) (H : i < (List.range' n m step).length) : (List.range' n m step)[i] = n + step * i

theorem List.head?_range' {s : Nat} (n : Nat) : (List.range' s n).head? = if n = 0 then none else some s

@[simp]

theorem List.head_range' {s : Nat} (n : Nat) (h : List.range' s n ≠ []) : (List.range' s n).head h = s

@[simp]

theorem List.map_add_range' (a s n step : Nat) : List.map (fun (x : Nat) => a + x) (List.range' s n step) = List.range' (a + s) n step

theorem List.range'_append (s m n step : Nat) : List.range' s m step ++ List.range' (s + step * m) n step = List.range' s (n + m) step

@[simp]

theorem List.range'_append_1 (s m n : Nat) : List.range' s m ++ List.range' (s + m) n = List.range' s (n + m)

theorem List.range'_sublist_right {step s m n : Nat} : (List.range' s m step).Sublist (List.range' s n step) ↔ m ≤ n

theorem List.range'_subset_right {step s m n : Nat} (step0 : 0 < step) : List.range' s m step ⊆ List.range' s n step ↔ m ≤ n

theorem List.range'_subset_right_1 {s m n : Nat} : List.range' s m ⊆ List.range' s n ↔ m ≤ n

theorem List.range'_concat {step : Nat} (s n : Nat) : List.range' s (n + 1) step = List.range' s n step ++ [s + step * n]

theorem List.range'_1_concat (s n : Nat) : List.range' s (n + 1) = List.range' s n ++ [s + n]

theorem List.range'_eq_cons_iff {s n a : Nat} {xs : List Nat} : List.range' s n = a :: xs ↔ s = a ∧ 0 < n ∧ xs = List.range' (a + 1) (n - 1)

range

theorem List.range_loop_range' (s n : Nat) : List.range.loop s (List.range' s n) = List.range' 0 (n + s)

theorem List.range_eq_range' (n : Nat) : List.range n = List.range' 0 n

theorem List.getElem?_range {m n : Nat} (h : m < n) : (List.range n)[m]? = some m

@[simp]

theorem List.getElem_range {n : Nat} (m : Nat) (h : m < (List.range n).length) : (List.range n)[m] = m

theorem List.range_succ_eq_map (n : Nat) : List.range (n + 1) = 0 :: List.map Nat.succ (List.range n)

theorem List.range'_eq_map_range (s n : Nat) : List.range' s n = List.map (fun (x : Nat) => s + x) (List.range n)

@[simp]

theorem List.length_range (n : Nat) : (List.range n).length = n

@[simp]

theorem List.range_eq_nil {n : Nat} : List.range n = [] ↔ n = 0

theorem List.range_ne_nil {n : Nat} : List.range n ≠ [] ↔ n ≠ 0

@[simp]

theorem List.tail_range (n : Nat) : (List.range n).tail = List.range' 1 (n - 1)

@[simp]

theorem List.range_sublist {m n : Nat} : (List.range m).Sublist (List.range n) ↔ m ≤ n

@[simp]

theorem List.range_subset {m n : Nat} : List.range m ⊆ List.range n ↔ m ≤ n

theorem List.range_succ (n : Nat) : List.range n.succ = List.range n ++ [n]

theorem List.range_add (a b : Nat) : List.range (a + b) = List.range a ++ List.map (fun (x : Nat) => a + x) (List.range b)

theorem List.head?_range (n : Nat) : (List.range n).head? = if n = 0 then none else some 0

@[simp]

theorem List.head_range (n : Nat) (h : List.range n ≠ []) : (List.range n).head h = 0

theorem List.getLast?_range (n : Nat) : (List.range n).getLast? = if n = 0 then none else some (n - 1)

@[simp]

theorem List.getLast_range (n : Nat) (h : List.range n ≠ []) : (List.range n).getLast h = n - 1

enumFrom

@[simp]

theorem List.enumFrom_eq_nil {α : Type u_1} {n : Nat} {l : List α} : List.enumFrom n l = [] ↔ l = []

@[simp]

theorem List.enumFrom_length {α : Type u_1} {n : Nat} {l : List α} : (List.enumFrom n l).length = l.length

@[simp]

theorem List.getElem?_enumFrom {α : Type u_1} (n : Nat) (l : List α) (m : Nat) : (List.enumFrom n l)[m]? = Option.map (fun (a : α) => (n + m, a)) l[m]?

@[simp]

theorem List.getElem_enumFrom {α : Type u_1} (l : List α) (n i : Nat) (h : i < (List.enumFrom n l).length) : (List.enumFrom n l)[i] = (n + i, l[i])

@[simp]

theorem List.tail_enumFrom {α : Type u_1} (l : List α) (n : Nat) : (List.enumFrom n l).tail = List.enumFrom (n + 1) l.tail

theorem List.map_fst_add_enumFrom_eq_enumFrom {α : Type u_1} (l : List α) (n k : Nat) : List.map (Prod.map (fun (x : Nat) => x + n) id) (List.enumFrom k l) = List.enumFrom (n + k) l

theorem List.map_fst_add_enum_eq_enumFrom {α : Type u_1} (l : List α) (n : Nat) : List.map (Prod.map (fun (x : Nat) => x + n) id) l.enum = List.enumFrom n l

theorem List.enumFrom_cons' {α : Type u_1} (n : Nat) (x : α) (xs : List α) : List.enumFrom n (x :: xs) = (n, x) :: List.map (Prod.map (fun (x : Nat) => x + 1) id) (List.enumFrom n xs)

@[simp]

theorem List.enumFrom_map_fst {α : Type u_1} (n : Nat) (l : List α) : List.map Prod.fst (List.enumFrom n l) = List.range' n l.length

@[simp]

theorem List.enumFrom_map_snd {α : Type u_1} (n : Nat) (l : List α) : List.map Prod.snd (List.enumFrom n l) = l

theorem List.enumFrom_eq_zip_range' {α : Type u_1} (l : List α) {n : Nat} : List.enumFrom n l = (List.range' n l.length).zip l

@[simp]

theorem List.unzip_enumFrom_eq_prod {α : Type u_1} (l : List α) {n : Nat} : (List.enumFrom n l).unzip = (List.range' n l.length, l)

enum

theorem List.enum_cons {α✝ : Type u_1} {a : α✝} {as : List α✝} : (a :: as).enum = (0, a) :: List.enumFrom 1 as

theorem List.enum_cons' {α : Type u_1} (x : α) (xs : List α) : (x :: xs).enum = (0, x) :: List.map (Prod.map (fun (x : Nat) => x + 1) id) xs.enum

theorem List.enum_eq_enumFrom {α : Type u_1} {l : List α} : l.enum = List.enumFrom 0 l

theorem List.enumFrom_eq_map_enum {α : Type u_1} (l : List α) (n : Nat) : List.enumFrom n l = List.map (Prod.map (fun (x : Nat) => x + n) id) l.enum